# SIMPLY SUPPORTED BEAM FORMULA

### Simply Supported Beam with Point Load Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=\frac { P(L-a) }{ L }$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { Pa }{ L }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-P{ \left< x-a \right> }^{ 0 }$$ Moment at distance x [M] $$M={ R }_{ 1 }x-P{ \left< x-a \right> }$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ Deflection at distance x [y] $$y={ \theta }_{ 1 }x+\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { P }{ 6EI } { \left< x-a \right> }^{ 3 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=\frac { -Pa }{ 6EIL } \left( 2L-a)(L-a \right)$$ Slope 2 [θ2] $${ \theta }_{ 2 }=\frac { Pa }{ 6EIL } ({ L }^{ 2 }-{ a }^{ 2 })$$ Slope [θ] $${ \theta }={ \theta }_{ 1 }+\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } -\frac { P }{ 2EI } { \left< x-a \right> }^{ 2 }$$

Note: In these formulas,  equations in brackets "< >" are singularity functions.

### Simply Supported Beam Distributed Load Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=\frac { { w }_{ a } }{ 2L } { (L-a) }^{ 2 }+\frac { { w }_{ L }-{ w }_{ a } }{ 6L } { (L-a) }^{ 2 }$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { { w }_{ a }+{ w }_{ L } }{ 2 } { (L-a) }-{ R }_{ 1 }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2\left( L-a \right) } \left< x-a \right> ^{ 2 }$$ Moment at distance x  [M] $$M={ R }_{ 1 }x-\frac { { w }_{ a } }{ 2 } { \left< x-a \right> }^{ 2 }-\frac { { w }_{ L }-{ w }_{ a } }{ 6(L-a) } { \left< x-a \right> }^{ 3 }$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ Deflection at distance x [y] $$y={ \theta }_{ 1 }x+\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } -\frac { { w }_{ a } }{ 24EI } { \left< x-a \right> }^{ 4 }-\frac { { w }_{ L }-{ w }_{ a } }{ 120EI(L-a) } { \left< x-a \right> }^{ 5 }$$ Shear force at distance x [V] $$V={ R }_{ 1 }-{ w }_{ a }\left< x-a \right> -\frac { { w }_{ L }-{ w }_{ a } }{ 2\left( L-a \right) } { \left< x-a \right> }^{ 2 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=\frac { -{ w }_{ a } }{ 24EIL } { (L-a) }^{ 2 }\cdot ({ L }^{ 2 }+2aL-{ a }^{ 2 })-\frac { { w }_{ L }-{ w }_{ a } }{ 360EIL } \cdot { (L-a) }^{ 2 }\cdot (7{ L }^{ 2 }-6al-3{ a }^{ 2 })$$ Slope 2 [θ2] $${ \theta }_{ 2 }=\frac { { w }_{ a } }{ 24EIL } { ({ L }^{ 2 }-{ a }^{ 2 }) }^{ 2 }+\frac { { w }_{ L }-{ w }_{ a } }{ 360EIL } { (L-a) }^{ 2 }(8{ L }^{ 2 }+9aL+3{ a }^{ 2 })$$ Slope [θ] $$\theta ={ \theta }_{ 1 }+\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } -\frac { { w }_{ a } }{ 6EI } { \left< x-a \right> }^{ 3 }-\frac { { w }_{ L }-{ w }_{ a } }{ 24EI(L-a) } { \left< x-a \right> }^{ 4 }$$

### Simply Supported Beam with Moment Formulas:

 Parameter Equation Reaction Force 1 [R1] $${ R }_{ 1 }=\frac { -{ M }_{ 0 } }{ L }$$ Reaction Force 2 [R2] $${ R }_{ 2 }=\frac { { M }_{ 0 } }{ L }$$ Shear force at distance x [V] $${ V=R }_{ 1 }$$ Moment at distance x  [M] $$M={ R }_{ 1 }x+{ M }_{ 0 }{ \left< x-a \right> }^{ 0 }$$ Bending stress at distance x [σ] $$σ=\frac { M\cdot c }{ I }$$ Deflection at distance x [y] $$y={ \theta }_{ 1 }x+\frac { { R }_{ 1 }{ x }^{ 3 } }{ 6EI } +\frac { M_{ 0 } }{ 2EI } { \left< x-a \right> }^{ 2 }$$ Slope 1 [θ1] $${ \theta }_{ 1 }=\frac { -{ M }_{ 0 } }{ 6EIL } (2{ L }^{ 2 }-6aL+3{ a }^{ 2 })$$ Slope 2 [θ2] $${ \theta }_{ 2 }=\frac { { M }_{ 0 } }{ 6EIL } ({ L }^{ 2 }-3{ a }^{ 2 })$$ Slope [θ] $${ \theta }={ \theta }_{ 1 }+\frac { { R }_{ 1 }{ x }^{ 2 } }{ 2EI } +\frac { { M }_{ 0 } }{ EI } \left< x-a \right>$$

Note: In these formulas,  equations in brackets "< >" are singularity functions. >" are singularity functions.